Typographical Errors

This is a list of typographical errors in Subsystems of Second Order Arithmetic, Stephen G. Simpson, Springer-Verlag, 1999, XIV + 445 pages. This list is current as of today, August 21, 2006. These errors will be corrected in Subsystems of Second Order Arithmetic, Second Edition, to be published by the Association for Symbolic Logic.

Proof of Theorem I.9.1, displayed formula, replace $\vert c-c_k\vert<2^n$ by
$\vert c-c_k\vert<2^{-n}$ .
Theorem I.10.3, in item 8, replace ``set of sentences'' by ``consistent set of sentences''.
Proof of Lemma III.2.5, the subscripts need to be repaired. The proof should be as follows:

We first consider the case of a finite product $\widehat A=\prod_{i=1}^m\widehat A_k$ . In this case, for each $j\in\mathbb{N}$ , let the smallest such that $m\cdot 2^{-l}\le 2^{-j}$ . Put $n_j=\prod_{k=1}^m(n_{l_jk}+1)-1$ and let $\langle x_{i j}: i\le n_j\rangle$ be an enumeration of $\prod_{k=1}^m\{x_{i l_j k}: i\le n_{l_j k}\}$ . Then $\langle\langle x_{i j}: i\le n_j\rangle: j\in\mathbb{N}\rangle$ attests to the compactness of $\widehat A$ .

In the case of a countably infinite product $\widehat A=\prod_{k\in\mathbb{N}}\widehat A_k$ , for each $j\in\mathbb{N}$ let smallest such that $(j+2)\cdot2^{-l}\le 2^{-j-1}$ . Put $n_j=\prod_{k=0}^{j+1} (n_{l_jk}+1)-1$ and let $\langle x_{i j}: i\le n_j\rangle$ be an enumeration of $\prod_{k=0}^{j+1}\{x_{i l_j k}: i\le n_{l_j k}\}$ . Again $\langle\langle x_{i j}: i\le n_j\rangle: j\in\mathbb{N}\rangle$ attests to the compactness of $\widehat A$ . This completes the proof of the lemma.
Proof of Theorem III.3.2, main paragraph, replace $h:K\to\mathbb{Q}$ by
$g:K\to\overline\mathbb{Q}$ , and replace by . Also, replace all three occurrences of by .
Proof of Theorem III.4.3, last line of third paragraph, replace
$\forall n (q_{2n}\ne0\to f(q_{2n+1}/q_{2n})=n)$
by
$\forall n (q_{2n}\ne0\to f(q_{2n+1}/q_{2n})=n)$ and $\forall n (q_{2n}=0\to q_{2n+1}=0)$ .
Proof of Theorem III.6.5, middle of page 120, replace $y_{ij}=p_{f(i)}z_{i,j+1}$ by $y_{ij}=p_{f(i)}y_{i,j+1}$ .
Proof of Theorem III.6.5, first line of last paragraph on page 120, replace $h_i:A\to D,i=1,2$ by $h_1,h_2:A\to D$ .
Proof of Lemma IV.1.4, end of proof, replace $f^{\textstyle{*}}(\sum^{j-1}_{i=0}f(i)+k)=1$ by
$f^{\textstyle{*}}(\sum^{j-1}_{i=0}g(i)+k)=1$ .
Exercise IV.2.10, replace ``closed set'' by ``closed set ''.
Definition V.4.5, replace A by .
Remark V.10.1, replace $\Sigma^0_1$ - $\mathsf{AC}_0$ by $\Sigma^1_1$ - $\mathsf{AC}_0$ .
Theorem VI.2.6, replace ``Over'' by ``over''.
Proof of Sublemma VI.3.3, in the definition of , replace $X_1\oplus X_1$ by $X_0\oplus X_1$ .
Definition VII.3.2.4, replace by .
Proof of Theorem VII.3.31, first line, replace T by .
Proof of Theorem VII.6.9, first paragraph, second last sentence, replace ``The'' by ``Then''.
Remark VIII.1.16, replace ``as do $\mathsf{ACA}_0$ and $\Sigma^1_1$ - $\mathsf{AC}_0$ '' by ``while $\mathsf{ACA}_0$ and $\Sigma^1_1$ - $\mathsf{AC}_0$ prove the same $\Pi^1_2$ sentences.''
Proof of Lemma VIII.2.16, first line of page 323, replace by .
Proof of Lemma IX.2.4, second line, replace $\vert M\vert\cup\mathcal{S}_{M'}$ by $\vert M\vert\cup\mathcal{S}_M$ .
Theorem X.2.9, items 5, 6, and 7, replace by $X^{\textstyle{*}}$ .
Section X.3, just before Definition X.3.1, replace ``Ramsey's theorem'' by ``Ramsey's theorem for exponent 3''.
Definition X.4.1, replace ``atomic formula'' by ``numerical term''.
Bibliography, item 268, replace ``borel'' by ``Borel''.
Index, atomic formula, replace ``2, 410'' by ``2''.
Index, formula, atomic, replace ``2, 410'' by ``2''.
Index, GKT basis theorem, replace ``325-326'' by ``325-326, 354-356''.
Index, numerical term, replace ``23'' by ``2, 23, 410''.
Index, term, numerical, replace ``2, 23'' by ``2, 23, 410''.
Index, add index entry, weak $\Sigma^1_1$ - $\mathsf{AC}_0$ , 342-347.
Index, universal $\Sigma^1_1$ , replace ``333, 356'' by ``252, 333, 356''.